MinT - Best Known (29, 29+29, s)-Nets in Base 4

Best Known (29, 29+29, s)-Nets in Base 4

(29, 29+29, 44)-Net over F₄ — Constructive and digital

Digital (29, 58, 44)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 19, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (10, 39, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]

(29, 29+29, 45)-Net in Base 4 — Constructive

(29, 58, 45)-net in base 4, using

2 times m-reduction [i] based on (29, 60, 45)-net in base 4, using
- base change [i] based on digital (9, 40, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]

(29, 29+29, 55)-Net over F₄ — Digital

Digital (29, 58, 55)-net over F₄, using

t-expansion [i] based on digital (26, 58, 55)-net over F₄, using
- net from sequence [i] based on digital (26, 54)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 26 and N(F) ≥ 55, using
    - Drinfeld modules of rank 1 [i]

(29, 29+29, 558)-Net in Base 4 — Upper bound on s

There is no (29, 58, 559)-net in base 4, because

1 times m-reduction [i] would yield (29, 57, 559)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 21009 458378 501477 472521 108203 730845 > 4⁵⁷ [i]